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## SAT

### Course: SAT>Unit 6

Lesson 6: Additional Topics in Math: lessons by skill

# Angles, arc lengths, and trig functions | Lesson

## What are "angles, arc lengths, and trig functions" problems, and how frequently do they appear on the test?

Note: On your official SAT, you'll likely see at most 1 question that tests your knowledge of the skills we teach in this article. Make sure you understand the more frequently-tested skills on the SAT before you spend time practicing this skill.

The problems in this lesson involve circles and angle measures in radians, a unit for angle measure much like degrees. We can use radian measures to calculate arc lengths and sector areas, and we can calculate the sine, cosine, and tangent of radian measures.
In this lesson, we'll learn to:
1. Convert between radians and degrees
2. Use our knowledge of special right triangles to find radian measures
3. Identify the sine, cosine, and tangent of common radian measures
This lesson builds upon the following skills:
You can learn anything. Let's do this!

## How do I convert between radians and degrees?

### Converting between radians and degrees

At the beginning of each SAT math section, the following information about radians and degrees is provided as reference:
• The number of degrees of arc in a circle is 360.
• The number of radians of arc in a circle is 2, pi.
This means 360 degrees is equivalent to 2, pi radians, and 180 degrees is equivalent to pi radians. We can set up a proportional relationship to convert between radian and degree measures.
start fraction, start text, r, a, d, i, a, n, space, m, e, a, s, u, r, e, end text, divided by, pi, end fraction, equals, start fraction, start text, d, e, g, r, e, e, space, m, e, a, s, u, r, e, end text, divided by, 180, degrees, end fraction

Example: Convert 90, degrees to radians.

This also means we can use radian measures to calculate arc lengths and sector areas just like we can with degree measures:
start fraction, start text, c, e, n, t, r, a, l, space, a, n, g, l, e, end text, divided by, 2, pi, end fraction, equals, start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, c, i, r, c, u, m, f, e, r, e, n, c, e, end text, end fraction, equals, start fraction, start text, s, e, c, t, o, r, space, a, r, e, a, end text, divided by, start text, c, i, r, c, l, e, space, a, r, e, a, end text, end fraction

Example: In a circle with center O, central angle A, O, B has a measure of start fraction, 2, pi, divided by, 3, end fraction radians. The area of the sector formed by central angle A, O, B is what fraction of the area of the circle?

### Try it!

try: compare radian and degree measures
Order the following angle measures from smallest to largest.

## How do I use special right triangles to find radian measures?

Note: The topics covered in this section have not appeared in recent SATs, but they could in the future! If they do, it will likely only be 1 question.

### Trig values of special angles

Trig values of π/4See video transcript

### Special right triangles in circles

At the beginning of each SAT math section, the following information about special right triangles is provided as reference:
These angle measures and their radian equivalents appear frequently in questions about circles and circle trigonometry. The table below shows the angles in special right triangles and their equivalent radian measures.
30, degreesstart fraction, pi, divided by, 6, end fraction
45, degreesstart fraction, pi, divided by, 4, end fraction
60, degreesstart fraction, pi, divided by, 3, end fraction
The radian measures we'll see on the SAT are usually multiples of the ones shown above.
On the test, we may be asked to find the radian measure of a central angle in a circle in the x, y-plane, such as that of angle A, O, B in the figure below. To do so, we'll draw a right triangle and look for the side length relationships in the special right triangles above.
We can draw a right triangle using the radius start overline, O, A, end overline as the hypotenuse. Since one vertex of the right triangle is the origin, the two legs of the right triangle have lengths equal to the x- and y- coordinates of point A.
Since the two legs of the right triangle have the same length, we can conclude that it is a 45-45-90 special right triangle, and the measure of angle A, O, B must be 45, degrees or start fraction, pi, divided by, 4, end fraction radians.

### Try it!

try: recognize a special right triangle in a circle
In the figure above, O is the center of a circle in the x, y-plane. The measure of angle A, O, B is start fraction, pi, divided by, 6, end fraction radians.
If the x-coordinate of point A is 2, square root of, 3, end square root, what is its y-coordinate?
What is the radius of the circle?

## How do I find the sine, cosine, and tangent of radian measures?

Note: The topics covered in this section have not appeared on recent tests, but they could show up on future tests! Questions on trigonometry in radians are a rare variety of an already infrequently-tested skill.

### The trig functions & right triangle trig ratios

The trig functions & right triangle trig ratiosSee video transcript

Trigonometry using radian measures is based on the unit circle, a circle centered on the origin with a radius of 1.
We can describe each point left parenthesis, x, comma, y, right parenthesis on the circle and the slope of any radius in terms of theta:
• x, equals, r, cosine, theta, equals, cosine, theta
• y, equals, r, sine, theta, equals, sine, theta
• start fraction, y, divided by, x, end fraction, equals, tangent, theta
The table below shows the sine, cosine, and tangent of some common radian measures in the unit circle:
Note: If you already know these, that's great! If not, consider spending time on the more frequently-tested skills on the SAT before familiarizing yourself with the values of trigonometric functions.
thetax or cosine, thetay or sine, thetatangent, theta
0100
start fraction, pi, divided by, 6, end fractionstart fraction, square root of, 3, end square root, divided by, 2, end fractionstart fraction, 1, divided by, 2, end fractionstart fraction, square root of, 3, end square root, divided by, 3, end fraction
start fraction, pi, divided by, 4, end fractionstart fraction, square root of, 2, end square root, divided by, 2, end fractionstart fraction, square root of, 2, end square root, divided by, 2, end fraction1
start fraction, pi, divided by, 3, end fractionstart fraction, 1, divided by, 2, end fractionstart fraction, square root of, 3, end square root, divided by, 2, end fractionsquare root of, 3, end square root
start fraction, pi, divided by, 2, end fraction01undefined
start fraction, 2, pi, divided by, 3, end fractionminus, start fraction, 1, divided by, 2, end fractionstart fraction, square root of, 3, end square root, divided by, 2, end fractionminus, square root of, 3, end square root
start fraction, 3, pi, divided by, 4, end fractionminus, start fraction, square root of, 2, end square root, divided by, 2, end fractionstart fraction, square root of, 2, end square root, divided by, 2, end fractionminus, 1
piminus, 100

The number of radians in a 135-degree angle can be written as a, pi, where a is a constant. What is the value of a ?

practice: use special right triangle to find radian measure
In the x, y-plane above, O is the center of the circle, and the measure of angle, A, O, B is start fraction, pi, divided by, a, end fraction. What is the value of a ?

## Things to remember

start fraction, start text, r, a, d, i, a, n, space, m, e, a, s, u, r, e, end text, divided by, pi, end fraction, equals, start fraction, start text, d, e, g, r, e, e, space, m, e, a, s, u, r, e, end text, divided by, 180, degrees, end fraction
We can describe each point left parenthesis, x, comma, y, right parenthesis on the unit circle and the slope of any radius in terms of theta:
• x, equals, cosine, theta
• y, equals, sine, theta
• start fraction, y, divided by, x, end fraction, equals, tangent, theta